4 1. Various Ways of Representing Surfaces and Examples

Figure 1.3. A torus and a handle.

Another familiar example of a surface is a torus—just as the

sphere is the surface of an idealised ball, the torus is the surface of

an idealised doughnut (or perhaps a bagel, depending on what sort

of diet one is on). Like our first three examples, it is a surface of

revolution, and may be obtained by rotating a circle around a line

which lies in the plane of the circle, but does not intersect it. We will

derive a nice equation (1.5) for the torus in the next lecture.

We can obtain new surfaces with qualitatively distinctive shapes

by the procedure called “attaching a handle”. A handle can be

thought of as a torus with a hole (or if you like, an inner tube with

a small patch cut out), as shown in Figure 1.3—this is attached to

a hole cut in a given surface. Applying this procedure to a sphere

produces a surface in the general shape of a torus. If we continue to

attach more handles, we obtain something reminiscent of a pretzel

with an increasing number of holes or, alternatively, a chain of tori

linked to each other—Figure 1.4 shows a sphere with two handles.

Like all the surfaces we have dealt with so far, these surfaces can

also be represented by equations with a certain amount of effort (see

Exercise 1.6).

b. Equations vs. other methods. We have obtained several dif-

ferent surfaces as the set of points whose coordinates (x, y, z) satisfy

one equation or another. It is natural to ask what sort of equations

will always yield nice, recognisable surfaces. Will any old equation

do? Or must we impose some restrictions? And conversely, can we

represent every surface by an equation?